Research article

Exact traveling wave solutions of the stochastic Wick-type fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation

  • Received: 06 November 2020 Accepted: 18 January 2021 Published: 04 February 2021
  • MSC : 35C07, 35K57

  • In this work, we test the intgrability of the stochastic Wick-type fractional Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGSK) equation on the Painlevé test and construct new Wick-type and nob-Wick-type versions of exact traveling wave solutions of the stochastic Wick-type fractional CDGSK equation by employing the Hermit transformation, the conformable fractional derivative and the sub-equations method. Moreover, we obtain exact traveling wave solutions of the fractional Sawada-Kotera (SK) equation and the fractional Caudrey-Dodd-Gibbon (CDG) equation as well. It is note that physical illustration may be useful to predict internal structure of the considered equations. The results confirm that sub-equations method is very effective and efficient to find exact traveling wave solutions of Wick-type fractional nonlinear evolution equations.

    Citation: Jin Hyuk Choi, Hyunsoo Kim. Exact traveling wave solutions of the stochastic Wick-type fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation[J]. AIMS Mathematics, 2021, 6(4): 4053-4072. doi: 10.3934/math.2021240

    Related Papers:

  • In this work, we test the intgrability of the stochastic Wick-type fractional Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGSK) equation on the Painlevé test and construct new Wick-type and nob-Wick-type versions of exact traveling wave solutions of the stochastic Wick-type fractional CDGSK equation by employing the Hermit transformation, the conformable fractional derivative and the sub-equations method. Moreover, we obtain exact traveling wave solutions of the fractional Sawada-Kotera (SK) equation and the fractional Caudrey-Dodd-Gibbon (CDG) equation as well. It is note that physical illustration may be useful to predict internal structure of the considered equations. The results confirm that sub-equations method is very effective and efficient to find exact traveling wave solutions of Wick-type fractional nonlinear evolution equations.



    加载中


    [1] D. Baleanu, M. Inc, A. Yusuf, A. I. Aliyu, Lie symmetry analysis, exact solutions and conservation laws for the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation, Commun. Nonlinear Sci. Numer. Simulat., 59 (2018), 222–234. doi: 10.1016/j.cnsns.2017.11.015
    [2] R. N. Aiyer, B. Fuchssteiner, W. Oevel, Solitons and discrete eigenfunctions of the recursion operator of non-linear evolution equaitons: the Caudrey-Dodd-Gibbon-Sawada-Kotera equations, J. Phys. A, 19 (1986), 3755–3770. doi: 10.1088/0305-4470/19/17/536
    [3] A. H. Salas, O. G. Hurtado, E. Jairo, E. Castillo, Computing multi-soliton solutions to Caudrey-Dodd-Gibbon equation by Hirota's method, Int. J. Phys Sci., 6 (2011), 7729–7737.
    [4] B. O. Jiang, B. Qinsheng, A study on the bilinear Caudrey-Dodd-Gibbon equation, Nonlinear Anal., 72 (2010), 4530–4533. doi: 10.1016/j.na.2010.02.030
    [5] N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311–328.
    [6] Y. Wang, L. Li, Lie Symmetry Analysis, Analytical Solution, and Conservation Laws of a Sixth-Order Generalized Time-Fractional Sawada-Kotera Equation, Symmetry, 11 (2019), 1436. doi: 10.3390/sym11121436
    [7] M. Safari, Application of He's Variational Iteration Method and Adomian Decomposition Method to Solution for the Fifth Order Caudrey-Dodd-Gibbon (CDG) Equation, Appl. Math., 02 (2011).
    [8] H. Naher, F. A. Abdullak, M. A. akbar, The $(G'/G)$-expansion method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon equation, Math. Problems Eng., 2011 (2011), ArticleID 218216.
    [9] M. Arshad, D. Lu, J. Wang, Abdullah, Exact traveling wave solutions of a fractional Sawada-Kotera Equation, East Asian J. Appl. Math., 8 (2018), 211–223. doi: 10.4208/eajam.090617.231117a
    [10] K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3–22. doi: 10.1023/A:1016592219341
    [11] K. Hosseini, M. Matinfar, M. Mirzazadeh, A (3+1)-dimentional resonant nonlinear Schrodinger equation and its Jacobi elliptic and exponential function solutions, Optik-Int. J. Light Elecron Optics, 207 (2020), 164458. doi: 10.1016/j.ijleo.2020.164458
    [12] K. Hosseini, M. Mirzazadeh, J. Vahidi, R. Asghari, Optical wave structures to the Fokas-Lenells equation, Optik-Int. J. Light Elecron Optics, 207 (2020), 164450. doi: 10.1016/j.ijleo.2020.164450
    [13] K. Hosseini, M. Mirzazadeh. J. F. Gomez-Aguilar, Soliton solutions of the Sasa-Satsuma equation in the monomode optical fibers including the beta-derivatives, Optik-Int. J. Light Elecron Optics, 224 (2020), 165425. doi: 10.1016/j.ijleo.2020.165425
    [14] K. Hosseini, M. Mirzazadeh, M. Ilie, J. F. Gomez-Aguilar, Biswas-Arshed equation with the beta time derivative: Optical solitons and other solutions, Optik-Int. J. Light Elecron Optics, 217 (2020), 164801. doi: 10.1016/j.ijleo.2020.164801
    [15] M. Wang, Y. Zhou. Z. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Physics Letter A, 216 (1996), 67–75.
    [16] A. M. Wazwaz, Multiple soliton solutions for (2+1)-dimensional Sawada-Kotera and Caudrey-Dodd-Gibbon equations, Math. Meth. Appl. Sci., 34 (2011), 1580–1586. doi: 10.1002/mma.1460
    [17] S. Bibi, N. Ahmed, I. Faisal, S. T. Mohyud-Din, M. Rafiq, U. Khan, Some new solutions of the Caudrey-Dodd-Gibbon (CDG) equation using the conformabel derivative, Adv. Differ. Equ., (2019), 2019: 89.
    [18] A. Neamaty, B. Agheli, R. Darzi, Exact traveling wave solutions for some nonlinear time fractional fifth-order Caudrey-Dodd-Gibbon equation by $(G'/G)$-expansion method, SeMA, 73 (2016), 121–129. doi: 10.1007/s40324-015-0059-4
    [19] N. A. Kudryashov, Painlevé analysis and exact solutions of the Korteweg-de Vries equation with a soure, Appl. Math. Lett., 41 (2015), 41–45. doi: 10.1016/j.aml.2014.10.015
    [20] R. Garrappa, E. Kaslik, M. Popolizio, Evaluation of fractional integrals and derivatives of elementary functions: Overview and tutorial, Mathematics, 407 (2019).
    [21] G. M. Bahaa, Fractional optimal control problem for differential system with control constraints, Filomat, 30 (2016), 2177-–2189. doi: 10.2298/FIL1608177B
    [22] M. K. Li, M. Sen, A. Pacheco-Vega, Fractional-Order-Based system identification for heat exchangers, proceedings of the 3rd world congress on momentum, Heat Mass Transfer, ENFHT, 134 (2018).
    [23] X. F. Pang, The Properties of the Solutions of Nonlinear Schrodinger Equation with Center Potential, Int. J. Nonlinear Sci. Numer. Simul., 15 (2014), 215–219.
    [24] B. Kafash, R. Lalehzari, A. Delavarkhalafi, E. Mahmoudi, Application of sochastic differential system in chemical reactions via simulation, MATCH Commun. Math. Comput. Chem., 71 (2014), 265–277.
    [25] C. H. Lee, P. Kim, An analyrical appreoach to solutions of master equations for stochastic nonlinear reactions, J. Math. Chem., 50 (2012), 1550–1569. doi: 10.1007/s10910-012-9988-7
    [26] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York, Wiley, 1993.
    [27] I. Podlubny, Fractional differential equations, San Diego, Academic Press, 1999.
    [28] B. Ahmad, S. K. Ntouyas, A. Alsaedi, On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions, Chaos Solitons Fract., 83 (2016), 234–241. doi: 10.1016/j.chaos.2015.12.014
    [29] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Theory Appl., 1993 (1993), Gordon and Breach, Yverdon, Switzerland.
    [30] M. Salinas, R. Salas, D. Mellado, A. Gları, C. Saavedra, A computational fractional signal derivative method, Model. Simul. Eng., 2018 (2018), Article ID 7280306.
    [31] Z. Korpinar, F. Tchier, M. Inc, F. Bousbahi, F. M. O. Tawfiq, M. A. Akinlar, Applicability of time conformable derivative to Wick-fractional-stochastic PDEs, Alexandria Eng. J., 59 (2020), 1485–1493. doi: 10.1016/j.aej.2020.05.001
    [32] J. H. Choi, H. Kim, R. Sakthivel, Periodic and solitary wave solutions of some important physical models with variable coefficients, Waves Random Complex Media, (2019), Available from: https://doi.org/10.1080/17455030.2019.1633029.
    [33] A. Atangana, R. T. Alqahtani, Modelling the spread of river blindness disease via the Caputo Fractional Derivative and the Beta-derivative, Entrophy, 18 (2016).
    [34] A. Atangana, E. F. D. Goufo, Extension of mathced asymtoic method to fractional boundary layers problems, Math. Problems Eng., 2014 (2014), 107535.
    [35] H. Holden, B. Øksendal, J. Uboe, T. Zhang, Stochastic partial differential equations, second edition, Universitext, Springer, New York, 2010.
    [36] M. J. Ablowitz, P. A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scattering, In: London Mathematical Society Lecture Note Series, 149, Cambridge University Press, 1996. 512.
    [37] M. J. Ablowitz, A. Ramani, H. Segar, A connection between nonlinear evolution equaions and ordinary differential equations of P-type. I, J. Math. Phys., 21 (1980), 715–721. doi: 10.1063/1.524491
    [38] M. J. Ablowitz, A. Ramani, H. Segar, A connection between nonlinear evolution equaions and ordinary differential equations of P-type. II, J. Math. Phys., 21 (1980), 1006–1015. doi: 10.1063/1.524548
    [39] F. Gao, C. Chi, Improvement on conformable fractional derivative and its applications in fractional differential equations, J. Function Spaces, 2020 (2020), 5852414. Available from: https://doi.org/10.1155/2020/5852414.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1922) PDF downloads(130) Cited by(2)

Article outline

Figures and Tables

Figures(9)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog